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Single TF

$\newcommand{\kon}{k_{\mathrm{on}}} \newcommand{\koff}{k_{\mathrm{off}}}$ <p>There exist two mathematical formalisms to describe the probabilities of molecular configurations at equilibrium: thermodynamics and kinetics. In the kinetics formalism, the system transits between configurational states according to rate parameters contained in differential equations, which can describe not just the system’s equilibrium but also its trajectory towards it, or the kinetics of non-equilibrium systems. In the thermodynamics/statistical mechanics formalism, we posit that a system will occupy configurations with lower energy, and use the Boltzmann distribution to estimate the proportion of time the system spends in each state, at equilibrium. The thermodynamics formalism is limited to describing equilibrium state probabilities, but it does so with fewer parameters.</p>

We’ll derive an expression for the probability of a single TFBS’ occupancy with both formalisms, but proceed with the thermodynamic description for more elaborate configurations. It will become clear why that is preferable.

Kinetics

Most derivations of the probability of single TFBS occupancy at equilibrium employ a kinetics formalism, so we’ll walk through that first, and then explore the analog in the thermodynamics description. In the kinetics description, the parameters are rates.

$\mathrm{TF} + \mathrm{TFBS} \underset{\koff}{\overset{\kon}{\rightleftarrows}} \mathrm{TF\colon TFBS}$

The natural rates are the rate of TF binding $\kon$ and unbinding $\koff$. Equilibrium is reached when binding and unbinding are balanced:

$\frac{d[\mathrm{TF\colon TFBS}]}{dt} = k_{\mathrm{on}}[\mathrm{TF}][\mathrm{TFBS}] - k_{\mathrm{off}}[\mathrm{TF\colon TFBS}] = 0 \text{ at equilibrium}$ $k_{\mathrm{on}}[\mathrm{TF}]_{\mathrm{eq}}[\mathrm{TFBS}]_{\mathrm{eq}} = k_{\mathrm{off}}[\mathrm{TF\colon TFBS}]_{\mathrm{eq}}$ $\text{(dropping eq subscript) }[\mathrm{TF\colon TFBS}] = \frac{k_{\mathrm{on}}[\mathrm{TF}][\mathrm{TFBS}]}{k_{\mathrm{off}}} = \frac{[\mathrm{TF}][\mathrm{TFBS}]}{k_{d}}$

where $k_{d} = \frac{\koff}{\kon}$ is called the dissociation constant (or equilibrium constant). We’d like to determine the probability of finding the TFBS occupied, i.e. the fraction of time it spends in the bound state. That fraction is $\frac{[\mathrm{bound}]}{([\mathrm{unbound}] + [\mathrm{bound}])} = \frac{[\mathrm{TF\colon TFBS}]}{([\mathrm{TFBS}] + [\mathrm{TF\colon TFBS}])}$. Define the denominator as $[\mathrm{TFBS}]{0} = [\mathrm{TFBS}] + [\mathrm{TF\colon TFBS}]$ so that $[\mathrm{TFBS}] = [\mathrm{TFBS}]{0} - [\mathrm{TF\colon TFBS}]$ and substitute:

$[\mathrm{TF\colon TFBS}] = \frac{[\mathrm{TF}]([\mathrm{TFBS}]_{0} - [\mathrm{TF\colon TFBS}])}{k_{d}}$ $[\mathrm{TF\colon TFBS}](k_d + [\mathrm{TF}]) = [\mathrm{TF}][\mathrm{TFBS}]_{0}$ $\frac{[\mathrm{TF\colon TFBS}]}{[\mathrm{TFBS}]_{0}} = \frac{[\mathrm{TF}]}{k_d + [\mathrm{TF}]}$ <div class="flash"> Note: We could also ask for this expression in terms of $[\mathrm{TF}]_0 = [\mathrm{TF}] + [\mathrm{TF\colon TFBS}]$ however, since we’re considering a single TFBS, $[\mathrm{TF\colon TFBS}]$ is at most 1, and so $[\mathrm{TF}]_0 \approx [\mathrm{TF}]$. In instances of ligand-receptor binding in which that approximation cannot be made, the fraction bound is a messy quadratic. Derivation here. </div>

Thermodynamics

In the thermodynamics description, the parameters are Gibbs free energies $\Delta G$. Let’s follow the derivation from Physical Biology of the Cell (pp. 242) and consider the number microstates underlying each of the of bound and unbound macrostates, and their energies.

In order to count microstates, we imagine distributing $L$ TF molecules across a space-filling lattice with $\Omega$ sites. The energy of a TF in solution is $\varepsilon_{\mathrm{solution}}$ and the energy of a bound TF is $\varepsilon_{\mathrm{bound}}$. $\beta$ is the constant $1/k_b T$ where $k_b$ is Boltzmann’s constant and $T$ is the temperature.

State	Energy	Multiplicity	Weight
	$A \cdot A_s$	$\frac{\Omega!}{(\Omega - A)!A!} \approx \frac{\Omega^{A}}{A!}$	$\frac{\Omega^{A}}{A!} \cdot e^{-\beta \left[ A \cdot A_s \right]}$
	$(A - 1) A_s + A_b$	$\frac{\Omega!}{(\Omega - (A - 1))!(A-1)!B!} \approx \frac{\Omega^{A-1}}{(A-1)!}$	$\frac{\Omega^{A-1}}{(A-1)!} \cdot e^{-\beta \left[ (A - 1) A_s + A_b \right]}$

In our case, the number of microstates in the unbound macrostate is $\frac{\Omega !}{L!(\Omega -L)!}\approx \frac{\Omega^L}{L!}$ and they each have energy $L \cdot \varepsilon_s$. The number of microstates in the bound macrostate is $\frac{\Omega !}{(L-1)!(\Omega -(L+1))}\approx \frac{\Omega^{(L-1)}}{(L-1)!}$ and they each have energy $(L-1) \varepsilon_s + \varepsilon_b$.

The Boltzmann distribution describes the probability of a microstate as a function of its energy: $p(E_i) = e^{-\beta E_i}/Z$ where $Z$ is the “partition function” or simply $\sum_i e^{-\beta E_i}$ the sum of the weights of the microstates, which normalizes the distribution. In our case:

$Z(L,\Omega)=\left(\colorbox{LightCyan}{$ \frac{\Omega^L}{L!} e^{-\beta L \varepsilon_s}$}\right) + \left(\colorbox{Seashell}{$\frac{\Omega^{(L-1)}}{(L-1)!} e^{-\beta [(L-1) \varepsilon_s + \varepsilon_b]}$}\right)$

With that expression in hand, we can express the probability of the bound macrostate, $p_b$:

$p_b=\frac{ \colorbox{Seashell}{$\frac{\Omega^{(L-1)}}{(L-1)!} e^{-\beta [(L-1) \varepsilon_s + \varepsilon_b]}$}}{\colorbox{LightCyan}{$\frac{\Omega^L}{L!} e^{-\beta L \varepsilon_s}$} + \colorbox{Seashell}{$\frac{\Omega^{(L-1)}}{(L-1)!} e^{-\beta [(L-1) \varepsilon_s + \varepsilon_b]}$}} \cdot \color{DarkRed}\frac{\frac{\Omega^L}{L!}e^{\beta L \varepsilon_s}}{\frac{\Omega^L}{L!}e^{\beta L \varepsilon_s}} \color{black} = \frac{(L/\Omega)e^{- \beta \Delta \varepsilon}}{1+(L/\Omega)e^{- \beta \Delta \varepsilon}}$

Where we have defined $\Delta \varepsilon = \varepsilon_b - \varepsilon_s$. $L/\Omega$ is really just a dimensionless TF concentration, which we’ll hand-wave as being equivalent to $[\mathrm{TF}]$, which leaves us with an expression suspiciously similar to the one we derived from the kinetics formalism:

$p_b = \frac{[\mathrm{TF}]e^{-\beta \Delta \varepsilon}}{1+[\mathrm{TF}]e^{-\beta \Delta \varepsilon}} \cdot \color{DarkRed}\frac{e^{\beta \Delta \varepsilon}}{e^{\beta \Delta \varepsilon}} \color{black} = \frac{[\mathrm{TF}]}{e^{\beta \Delta \varepsilon}+[\mathrm{TF}]}$

From which we recapitulate an important correspondence between kinetics and thermodynamics at equilibrium: $ k_d = e^{\beta \Delta \varepsilon} = e^{\Delta \varepsilon / k_bT} $ more commonly written for different units as $k = e^{-\Delta G / RT}$.

The takeaway is that both the kinetics and thermodynamics formalisms produce an equivalent expression for the probabilities of each of the bound and unbound configurations, parameterized respectively by $k_d$ and $\Delta G$.

References:

Sample Values

In order to compute probabilities like $p_b$, we need concrete TF concentrations $[\mathrm{TF}]$ and binding affinities (either $k_d$ or $\Delta G$). What are typical intranuclear TF concentrations and binding affinities?

Concentrations

A typical human cell line, K562s, have a cellular diameter of 17 microns. (BioNumbers)

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K562 volume: 2572 μm^3

A typical expressed TF has a per-cell copy number range from $10^3$ - $10^6$. (BioNumbers)

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If TF copy numbers range from 1,000-1,000,000, then TF concentrations range from 1nM to 646nM

We might also like a distribution over this range. Let’s posit a lognormal, where $10^3$ and $10^6$ are the 3σ from the mean, which is $10^{4.5}$. Then $\sigma = 10^{0.5}$

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Affinities

What are typical TF ΔG’s of binding? How about the $\koff$ rates and half lives?

We can use the prior knowledge that dissociation constants should be in the nanomolar regime (BioNumbers).
We can use the relation that $\Delta G = -k_b T \cdot \ln(k_d)$ (Plugging in 310°K (human body temp) and the Boltzmann constant $k_b$ in kcal/Mol)
We use the approximation that $\kon$ is ~$10^5 / $ Molar $ \times $ sec (Wittrup)

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We learn that an order of magnitude residence time difference results from just 1.4 extra kcal/Mol, and that TF half lives range from about 5s to about 2h. Let’s once again posit a distribution of affinities to sample from (defined on $k_d$):

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With those concrete TF concentrations and dissociation constants, we can finally plot our function $p_b = \frac{[\mathrm{TF}]}{e^{\beta \Delta \varepsilon}+[\mathrm{TF}]}$.

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(Note that both $[\mathrm{TF}]$ and $k_d$ are plotted in log space, but $p_b$ is linear.)

Multiple TFs: Direct Cooperativity & Competition

Suppose now that two TFs bind adjacent segments of DNA in such a way that the binding of either facilitates (or hinders) the binding of the other. We call this direct cooperativity (and competition). We’ll focus first on cooperativity.

Cooperativity

As before, the statistical mechanics formalism entails enumerating the configurations, their multiplicities, and their energies. We’ll call the TFs A and B. We’ll denote their counts as $A$ and $B$. The energy of a TF in solution will once more be $A_s$ and bound to its cognate TFBS $A_b$. The energy of cooperativity will be $C_{AB}$.

State	Energy	Multiplicity	Weight
	$A \cdot A_s + B \cdot B_s$	$\frac{\Omega!}{(\Omega - A - B)!A!B!} \approx \frac{\Omega^{A+B}}{A!B!}$	$\frac{\Omega^{A+B}}{A!B!} \cdot e^{-\beta \left[ A \cdot A_s + B \cdot B_s \right]}$
	$(A - 1) A_s + A_b + B \cdot B_s$	$\frac{\Omega!}{(\Omega - (A - 1) - B)!(A-1)!B!} \approx \frac{\Omega^{A+B-1}}{(A-1)!B!}$	$\frac{\Omega^{A+B-1}}{(A-1)!B!} \cdot e^{-\beta \left[ (A - 1) A_s + A_b + B \cdot B_s \right]}$
	$A \cdot A_s + (B - 1) B_s + B_b$	$\frac{\Omega!}{(\Omega - A - (B - 1))!A!(B-1)!} \approx \frac{\Omega^{A+B-1}}{A!(B-1)!}$	$\frac{\Omega^{A+B-1}}{A!(B-1)!} \cdot e^{-\beta \left[ A \cdot A_s + (B - 1) B_s + B_b \right]}$
	$(A - 1) A_s + A_b + (B - 1) B_s + B_b + C_{AB}$	$\frac{\Omega!}{(\Omega - (A - 1) - (B-1))!(A-1)!(B-1)!} \approx \frac{\Omega^{A+B-2}}{(A-1)!(B-1)!}$	$\frac{\Omega^{A+B-2}}{(A-1)!(B-1)!} \cdot e^{-\beta \left[ (A - 1) A_s + A_b + (B - 1) B_s + B_b + C_{AB} \right]}$

The partition function is the sum of the weights:

$Z = \frac{\Omega^{A+B}}{A!B!} \cdot e^{-\beta \left[ A \cdot A_s + B \cdot B_s \right]} + \frac{\Omega^{A+B-1}}{(A-1)!B!} \cdot e^{-\beta \left[ (A - 1) A_s + A_b + B \cdot B_s \right]} + \frac{\Omega^{A+B-1}}{A!(B-1)!} \cdot e^{-\beta \left[ A \cdot A_s + (B - 1) B_s + B_b \right]} + \frac{\Omega^{A+B-2}}{(A-1)!(B-1)!} \cdot e^{-\beta \left[ (A - 1) A_s + A_b + (B - 1) B_s + B_b + C_{AB} \right]}$ <p>Which we can greatly simplify by multiplying the entire expression by the reciprocal of the “base state” weight, $\color{DarkRed}\frac{A!B!}{\Omega^{A+B}} \cdot e^{\beta \left[ A \cdot A_s + B \cdot B_s \right]}$, normalizing that weight to 1:</p> $Z = 1 + \frac{A}{\Omega} \cdot e^{-\beta \left[ A_b-A_s \right]} + \frac{B}{\Omega} \cdot e^{-\beta \left[ B_b-B_s \right]} + \frac{A \cdot B}{\Omega^2} \cdot e^{-\beta \left[ A_b-A_s+B_b-B_s+C_{AB} \right]}$ <p>Taking the definition $[A] = A/\Omega$ and $\Delta G_A = A_b-A_s$ produces:</p> $Z = 1 + [A] e^{-\beta \left[ \Delta G_A \right]} + [B] e^{-\beta \left[ \Delta G_B \right]} + [A][B] e^{-\beta \left[ \Delta G_A+\Delta G_B+C_{AB} \right]}$ <p>Then the probability of any state is just the weight of that state (scaled by the weight of the base state) divided by the partition function expression $Z$.</p>

From the above, we notice the form of the expression for the weight of a configuration of N TFBSs:

$p_{\mathrm{config}} = \prod_{i \in \, \mathrm{bound \,TBFS}} \left( [\mathrm{TF}_{\mathrm{cognate}(i)}] \cdot e^{-\beta \left[ \Delta G_i + \sum_j c_{ij} \right]} \right) / Z$

For numerical stability, we take the log of the unnormalized probability (that is, the weight) of configurations:

$\log(\tilde{p}_{\mathrm{config}}) = \sum_{i \in \, \mathrm{bound \,TBFS}} \left( \log([\mathrm{TF}_{\mathrm{cognate}(i)}]) - \beta \left[ \Delta G_i + \sum_j c_{ij} \right] \right)$

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Competition

Incorporating competition into our thermodynamic model is slightly more subtle than cooperativity, because we can imagine two types of competition

Two TFs which may both bind DNA at adjacent sites, causing a free energy penalty due to some unfavorable interaction.
Two TFs with overlapping DNA binding sites, which cannot physically be bound at the same time.

In the former case, the expression for $p_\mathrm{config}$ we had written before suffices, with values of $C_{AB}$ having both signs to represent cooperativity and competition. Nominally, the latter case also fits this formalism if we allow $C_{AB}$ to reach $-\infty$, but that would cause us headaches in the implementation. Instead, the weights of all those configurations which are not physical attainable, due to “strict” competitive binding between TFs vying for overlapping binding sites, are merely omitted from the denominator $Z$.

References:

Sample cooperativity values

In order to compute concrete probabilities of configurations, accounting for cooperativity and competition, we will need concrete energies. We’ll take $C_{AB}$ to be distributed exponentially with a mean at 2.2kcal/Mol. (Forsén & Linse).

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Let’s check our derivation (and our implementation) of $p_\mathrm{config}$ by comparing it to our direct computation of $p_b$ from §1. Single TF.

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p_bound (prev method): 	 0.00921878699097193
p_config (new method): 	 [(array([0]), 0.9907812130090282), (array([1]), 0.00921878699097194)]

With that sanity check, let’s now consider the scenario of two transcription factors with cooperative binding, and compute the probabilities of each of the 4 configurations:

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Low-Affinity binding

In reality, transcription factors’ binding sites are not so discretely present or absent: transcription factors may bind anywhere along the DNA polymer, with an affinity dependent on the interaction surface provided by the sequence of nucleotides at each locus. There are a variety of approaches to model the sequence-to-affinity function for each TF. The simplest is to consider each nucleotide independently, and list the preferences for each nucleotide at each sequential position in a matrix. This type of “mono-nucleotide position weight matrix” (PWM) model is commonly used, and frequently represented by a “sequence logo” plot.

PWMs: Construction

I have curated PWMs from various databases for ~1000 human TFs.

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As an example, consider the Transcription Factor SPI1 / PU.1, for which the following PWMs are registered in the CisBP 2.00 database:

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This representation of this matrix visualizes the proportion of occurrences of a base at a particular position in aligned SPI1 binding sites. Since they look similar, we may be tempted to average them to have a single model of PU.1 binding, however it is considered unwise to do so, as TFs are believed to have multiple binding “modes” which refers explicitly to modes in the distribution of sequence-binding affinities.

These PWM models can be fit from a variety of experiments measuring PU.1 binding. In this case, we’re looking at models fit from High-Throughput SELEX data, but other models for PU.1 binding generated from other types of experiments are catalogued at its entry in the HumanTFs database.

PWMs: scoring sequences

If we assume a TF’s PWM catalogues its relative binding preferences to every possible sequence of individual bases, we can use that PWM to evaluate the relative probabilities of TF binding to new sequences. We’ll define the PWM “score” as the (log) ratio of the probability of the sequence with the PWM against random chance. The probability of a sequence under the PWM is $\prod \mathrm{PWM}[i][\mathrm{dna}[i]]$. The probability of the sequence under a background model is $\prod \mathrm{bg}(\mathrm{dna}[i])$ where $\mathrm{bg}$ describes the frequency of each base in the genome (not quite 1/4 in the human genome). We take the logarithm of the ratio of these two probabilities to produce the log-likelihood ratio, which we call our “score”:

$\mathrm{score}(\mathrm{PWM}, j) = \log(\frac{\prod \mathrm{PWM}[i][\mathrm{dna}[j+i]]}{\prod \mathrm{bg}(\mathrm{dna}[j+i])}) = \sum \log(\frac{ \mathrm{PWM}[i][\mathrm{dna}[j+i]]}{\mathrm{bg}(\mathrm{dna}[j+i])})$

A positive $\mathrm{score}(\mathrm{PWM}, j)$ indicates the subsequence starting at position $j$ is likelier under the PWM model than a random sequence from the human genome, and a negative score indicates the subsequence is more likely to be random. Let’s visualize a PU.1 scoring matrix:

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Our strategy will be to score genomic regions for TF binding, searching for high PWM scores starting at every offset $j$ in the supplied DNA sequence. Let’s remind ourselves here that in order to find the full set of motif matches in a supplied DNA sequence, we either need to scan the reference genome and its reverse complement, or scan the positive strand with both the forward and reverse-complement PWMs. We opt for the latter approach, and therefore augment our PWM set with reverse-complement PWMs:

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Since we now have sequence-dependent transcription factor binding models, we will select a sequence which harbors known binding sites for select transcription factors, and see whether the models predict binding of those transcription factors.

As a specific sequence, let’s use the SPI1 promoter.

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TTCCAGGGAGGAAACCCTGACTTCCCACTGATAGCAAGCCAGGAGGGCAGTGGGTGGGCTGGCGGGTTCGTGGGCAGGCAGGCAGGCGTCCGAGGGCCACGGGTTGGGCTGGTGGAGGAGTCCCGGTACTCACAGGGGGGACGAGGGGAAACCCTTCCATTTTGCACGCCTGTAACATCCAGCCGGGCTCCGAGTCGGTCAGATCCCCTGCCTCGGTGGGGGCCAATGCAGAGCCCCTCAGGATGGGGTGCCCCGTCAGGGGCTGGACGGTCGTGGGGCGGGTGCAGGGCTCAGGCCTGCCCCCTGAGCTACAGGAGCCCTGGGTGAGCCCCCTCCCTTGACATTGCAGGGCCAGCACAAGTTCCTGATTTTATCGAAGGGCCTGCCGCTGGGAGATAGTCCCCTTGGGGTGACATCACCGCCCCAACCCGTTTGCATAAATCTCTTGCGCTACATACAGGAAGTCTCTGGCCGGCTGGGGCAGGTGGTGCTCAAAGGGCTGGCCTGGGAAGCCATGGGGTCCAGGCCCCCTGCCCAGAGGAAGCTGGGACTGAGAGGGATGACTTTGGGGGCTAAGCTGGGGAGGGAGGATGGGAGGGAGAACGTGTAGCTCTGCCACACCACTGGGAGGCTTTTGCTCTAACCCAACAAATGCCTGCTTCTTTTGAGATCCCTATGTAGCCAACAGTCACCTCATTGGGGTCAGAGCTGGAAGGGGTGGCCTCTTGGGGCCTCCACTTTCTGGAGTCAGCCTTCCTGGGTGAGGGCTCTGATCTAGCAGGCTATCAGGCCTGGCTCTTC

SPI1 promotes its own transcription, by binding to its own promoter. We can scan the SPI1 promoter for significant SPI1 PWM scores to locate the binding site.

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Now we’d like to score every PWM against every offset in the entire reference human genome. In order to accomplish that, we’ll use MOODS, which implements a faster scanning algorithm in C++. We’ll use this script to score the human reference genome with our PWMs.

PWMs: Score cutoffs

Once we have scanned a sequence of genomic DNA, scoring the PWM at every offset, we need to establish which of those scores is “significant” enough that we would predict the existence of a transcription factor binding site at that position in the genome. We can imagine two approaches to distinguish “significant” motif matches for each PWM. Thresholding on a particular score corresponds to predicting the TF is bound to sites which are $e^{\mathrm{score}}$ more likely under the PWM than random. A score cutoff of 6.91 is 1000x more likely under the PWM than random, for every PWM.

Contrarily, if you wanted to choose the 0.1% top scores for each PWM, each PWM would have a unique score cutoff, as each cutoff would depend on the distribution of scores specified by the PWM. At first this may seem surprising, so we’ll explore it empirically for two PWMs:

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Clearly we can see these two PWMs define different score distributions for random sequences. Let’s see what probabilities the highest scores correspond to for each of these PWMs, using MOODS’ score cutoff method:

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For each motif, we can compute which score corresponds to each p-value cutoff, and then estimate p-values for scores of motif matches discovered during scanning using an interpolation. We’ll use a generic spline as an interpolation function.

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PWMs: Other formalisms

Unfortunately, the PWM formalism we have introduced misses the biophysical interpretation we have carried throughout our analysis thus far. We can coax a biophysical interpretation from these PWMs as follows (following the approach from Foat, Morozov, Bussemaker, 2006):

If we consider the nucleotides independently, then the ratio of equilibrium binding to a given nucleotide $j$ versus the highest-affinity nucleotide $\mathrm{best}$ at each position is related to the difference in energies of binding:

$\frac{[TF:N_j]}{[TF:N_{best}]} = \frac{K_{a_j}}{K_{a_{best}}} = \exp \{\Delta G_j - \Delta G_{best} \}$

With this expression, we can compute the energy of binding of a TF to a sequence relative to that TF’s energy of binding to its best sequence of nucleotides. However, we need to know an absolute energy of binding to anchor this relative measure.

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We can visualize this “energy PWM” as well. Visualizing energies of particular bases relative to SPI1’s best sequence does not produce a visual representation that’s easy to reason about:

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We can rescale this matrix to the average sequence:

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However, the ideal rescaling of this PWM would indicate whether each nucleotide contributes positively or negatively towards the energy of binding. After all, SPI1 likely doesn’t bind the “average” sequence – which is to say it’s off rate is higher than it’s on rate.

We chose SPI1 / PU.1 as our prototype in part because its absolute binding energies to various sequences have been measured, allowing us to anchor our relative energies PWM model. Pham et al measured PU.1 binding to a variety of high-affinity sequences by microscale thermophoresis and recorded a $k_d$ of 156nm for the best one.

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9.648885505793956

However, the question remains of how to allocate those ~9.6 kcal/mol across the 14 bases of the SPI1 motif. The naive option is to uniformly spread the binding energy across the bases (9.6/14). Instead, we’ll distribute it proportionally to the variance at each position to capture the intuition that the flanking nucleotides contribute less.

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At last we have a visual representation of of a mono-nucleotide sequence preference model which captures our intuitions about the energetics of TF binding.

For a restricted set of TFs with high-quality HT-SELEX data, Rube et al produced relative energy PWMs with higher accuracy in the low-affinity regime. SPI1 is among that privileged set, so we’ll take a look at that PWM for comparison.

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PWMs: Evaluation

To what extent do these PWM models predict real transcription factor binding? In order to evaluate these models, we would like to compare their predictions with ground truth measurements of transcription factor occupancy.

TF occupancy can be estimated with ChIP. Has been estimated in K562 (and HepG2)

Limitations:

We only have PWMs for 1000 TFs,
We only have ChIP for 600 TFs

What’s the intersection?

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In ATAC OCRs

Justify only looking inside ATAC peaks / OCRs.

explain motivation: want to look in all ATAC regions for PWMs and chip signal in k562s

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from this we conclude that the first set of ATAC peaks is a richer dataset, and proceed with those

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Get DGF

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<AxesSubplot:xlabel='K562-DS15363_posterior', ylabel='K562-DS16924_posterior'>

Load K562 RNA

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(Marinov… Wold, 2014)

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(Li,… Biggin, 2014)

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<AxesSubplot:ylabel='Frequency'>

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Evaluation of PWMs at the SPI1 promoter

We’ve just generated a list of OCRs we want to evaluate PWMs for, but let’s start with our single example to determine how to evaluate PWMs vs ChIP.

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Get TF ChIP Peaks at SPI1 Promoter

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Get TF ChIP signal over background at SPI1 promoter

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Get ATAC signal over background at SPI1 promoter

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Get PWM predictions at SPI1 promoter

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3D Ridgeline

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genome_linspace: (800, 20)
signal_heights:	 (800, 20)
TF_y_index: 	 (20,)

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---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Input In [156], in <cell line: 23>()
     21 ax.set_ylim3d(TF_y_index.min(), TF_y_index.max())
     22 ax.set_ylabel('TF')
---> 23 ax.set_zlim3d(signal_heights.min(), signal_heights.max())
     24 ax.set_zlabel('signal fold change over background')
     26 xtick_vals = ax.get_xticks()

File ~/miniconda3/envs/py39/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:865, in Axes3D.set_zlim3d(self, bottom, top, emit, auto, zmin, zmax)
    862     top = zmax
    864 self._process_unit_info([("z", (bottom, top))], convert=False)
--> 865 bottom = self._validate_converted_limits(bottom, self.convert_zunits)
    866 top = self._validate_converted_limits(top, self.convert_zunits)
    868 old_bottom, old_top = self.get_zlim()

File ~/miniconda3/envs/py39/lib/python3.9/site-packages/matplotlib/axes/_base.py:3614, in _AxesBase._validate_converted_limits(self, limit, convert)
   3611 converted_limit = convert(limit)
   3612 if (isinstance(converted_limit, Real)
   3613         and not np.isfinite(converted_limit)):
-> 3614     raise ValueError("Axis limits cannot be NaN or Inf")
   3615 return converted_limit

ValueError: Axis limits cannot be NaN or Inf

2D Ridgeline

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Scatterplots

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name
ATF1        278.59146
ATF2        671.55342
ATF3       2017.85285
ATF4         61.95164
ATF7       1206.30707
              ...    
ZKSCAN1      28.54167
ZNF175       50.23833
ZNF281      112.63834
ZNF282       31.21194
ZNF766       56.29987
Name: max_chip_peak_height, Length: 87, dtype: float64

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TFAP2A     0.082112
TFAP2B     0.019229
TFAP2C     0.024566
TFAP2D     0.000000
TFAP2E     0.029581
            ...    
MTERF1     8.131813
MTERF2    12.372807
TP53       2.256278
TP63       0.674158
TP73       0.592430
Name: TF_RNA, Length: 1635, dtype: float64

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(0.07881793222722186, 0.06831818983387146)

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Metrics

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Generate reports for each peak

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<seaborn.axisgrid.JointGrid at 0x1b4ffe350>

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Evaluating across all ATAC peaks

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<AxesSubplot:xlabel='atac_signalValue', ylabel='num_chip_peaks'>

Per TF, Motif

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TFAP2A     0.082112
TFAP2B     0.019229
TFAP2C     0.024566
TFAP2D     0.000000
TFAP2E     0.029581
            ...    
MTERF1     8.131813
MTERF2    12.372807
TP53       2.256278
TP63       0.674158
TP73       0.592430
Name: TF_RNA, Length: 1635, dtype: float64

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225

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0    67053
dtype: int64

Probably no longer relevant:

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Let’s now proceed to a slightly more complex regulatory region, with 20 binding sites for 10 TFs

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Notice first that when binding sites overlap slightly, we’ve introduced a free energy penalty of competition, denoted by a pink line between the binding sites, and when they stronly overlap, we’ve disallowed those configurations, denoted by a red line.

We notice a problem with evaluating our $p_\mathrm{config}$ expression with realistic regulatory DNA: the number of configurations grows exponentially in the number of transcription factor binding sites ($2^{|\mathrm{TFBS}|}$). Only 20 Transcription Factor Binding sites entails ~1 million configurations, and so ~1 million terms in the denominator $Z$. Computing probabilities of configurations exactly then becomes intractable for realistic scenarios: we need to proceed by sampling.

Thankfully there’s a dynamic programming approach to computing Z.

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Which we can plot, and compute the associated log-statistical-weight:

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log(p_config) = -1.4748275325663975

References

Calculating transcription factor binding maps for chromatin

Multiple TFs: Nucleosome-mediated Cooperativity

nucleosomes as a soft AND gate

Direct cooperativity (driven by protein-protein interactions) between transcription factors exerts significant control over transcription when it occurs, but it appears to be the exception, not the rule. More common is indirect cooperativity mediated by nucleosomes. Nucleosomes can be thought of as repressive transcription factors with large footprints and relatively non-specific sequence-dependent binding affinities. When multiple transcription factors bind independently in the footprint of a nucleosome, they do so cooperatively, despite not making physical contact with one another.

Including nucleosomes in our description of chromatin produces a new set of interactions, which require energy parameters. Those are:

Nucleosome:DNA interactions
Nucleosome:TF interactions
Nucleosome:Nucleosome interactions

Nucleosome:DNA

Nucleosomes have at least a 5000-fold range of affinities for differing DNA sequences.

What controls nucleosome positions? Nucleosomes do not bind specific 147-bp sequences, but they do have preferences for certain periodic patterns, and disfavor certain other patterns. Nucleosomes are canonically wrapped by 147bp of DNA, but due to nucleosome crowding and nucleosome beathing, they are often wrapped by fewer bases.

For our model, we will say that any stretch of DNA [of lengths 100-147bp] can be wrapped by a nucleosome, with an energy which is derived from the [sequence model here].

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Nucleosome:TF

Nucleosomes and Transcription Factors were considered to sterically block one another from binding the same stretch of DNA, however, we now know that many TFs can co-bind DNA and stabilize nucleosomal binding to a piece of DNA.

For simplicity, we’ll assume we’re only considering transcription factors which cannot co-bind DNA with nucleosomes, and impose a large ΔG penalty for nucleosome and TF co-binding.

Nucleosome:Nucleosome

Nucleosomes are spaced by stretches of free DNA called linker DNA, which have characterisitic lengths. Those specific lengths are believed to facilitate nucleosome arrays forming higher-order chromatin structures.

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0.04610959744808222

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557.3903692694596

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send blog post for feedback:

Jeremy / Leonid
Advait / Bin Zhang
Anders
Kellis lab
Vlad Teif, other online nucleosome people
TF people?
Muir Morrison / Rob Philips
Bruce Tidor?

transfer matrix: probably won’t use

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12757

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TFAP2A    [[-0.5324, 0.34450000000000003, -1.5585, 0.530...
TFAP2B    [[-2.4892, 0.5962000000000001, 1.0115, -2.6127...
TFAP2C    [[-0.5975, -0.163, -1.1323, 0.7243], [-1.3232,...
TFAP2D    [[0.8455, -1.4861, 0.1063, -1.8089], [-5.6699,...
TFAP2E    [[-5.6699, 0.43660000000000004, 1.1706, -5.669...
                                ...                        
BATF3     [[-0.8147000000000001, 0.11900000000000001, 0....
CREB1     [[-0.049100000000000005, -0.20500000000000002,...
TP53      [[0.1729, -0.6791, 0.6519, -0.661], [0.7944, -...
TP63      [[-0.5391, -1.3228, 1.0218, -0.394300000000000...
TP73      [[0.08610000000000001, -0.6413, 0.738200000000...
Name: log_pwm, Length: 1080, dtype: object

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----------------------------------------------------------
IndexError               Traceback (most recent call last)
<ipython-input-231-ffe1fe50396a> in <module>
      4         i = transfer_matrix_index(TF, offset)
      5 
----> 6         transfer_matrix_a[i][i+1] = pwm[offset][base_index['a']]
      7 

IndexError: index 12757 is out of bounds for axis 0 with size 12757

Verify against proteomics

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passed 12755 symbols
31 names contained whitespace. Stripping...
558 duplicates. 12197 uniques.

querying 1-1000...done.
querying 1001-2000...done.
querying 2001-3000...done.
querying 3001-4000...done.
querying 4001-5000...done.
querying 5001-6000...done.
querying 6001-7000...done.
querying 7001-8000...done.
querying 8001-9000...done.
querying 9001-10000...done.
querying 10001-11000...done.
querying 11001-12000...done.
querying 12001-12197...done.
Finished.
728 input query terms found dup hits:
	[('ARX', 2), ('NCL', 2), ('CNP', 2), ('RPF1', 2), ('HYPK', 2), ('FH', 3), ('RPL17', 3), ('ATF2', 2),
13 input query terms found no hit:
	['FLJ22184', 'L1RE1', 'MT-ND5', 'MT-ND4', 'MT-CO1', 'MT-CYB', 'MT-ND1', 'GAGE3', 'MT-ATP8', 'hCG_198

unchanged: 11537;  updates: 647;  missing: 13

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0.387037037037037

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TFAP2A   -2.802736
TFAP2B   -2.548439
TFAP2C   -3.410715
TFAP2D         NaN
TFAP2E         NaN
            ...   
BATF3          NaN
CREB1     0.484013
TP53     -3.214319
TP63     -0.861330
TP73           NaN
Name: Gygi_K562, Length: 1080, dtype: float64

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0.060758414997869624

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passed 30872 symbols
2285 duplicates. 28587 uniques.

querying 1-1000...done.
querying 1001-2000...done.
querying 2001-3000...done.
querying 3001-4000...done.
querying 4001-5000...done.
querying 5001-6000...done.
querying 6001-7000...done.
querying 7001-8000...done.
querying 8001-9000...done.
querying 9001-10000...done.
querying 10001-11000...done.
querying 11001-12000...done.
querying 12001-13000...done.
querying 13001-14000...done.
querying 14001-15000...done.
querying 15001-16000...done.
querying 16001-17000...done.
querying 17001-18000...done.
querying 18001-19000...done.
querying 19001-20000...done.
querying 20001-21000...done.
querying 21001-22000...done.
querying 22001-23000...done.
querying 23001-24000...done.
querying 24001-25000...done.
querying 25001-26000...done.
querying 26001-27000...done.
querying 27001-28000...done.
querying 28001-28587...done.
Finished.
1899 input query terms found dup hits:
	[('GJC1', 2), ('ATP6C', 2), ('A15', 2), ('MYPOP', 2), ('LAG1', 4), ('LASS1', 2), ('UOG1', 2), ('SDP1
8595 input query terms found no hit:
	['FAM1B', 'DKFZp686P0738', 'hCG_23354', 'KIAA0693', 'HSPC198', 'BM-004', 'NKD', 'PP7246', 'hCG_18772

unchanged: 9507;  updates: 10576;  missing: 8595

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0.11231359181934385

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TFAP2A   NaN
TFAP2B   NaN
TFAP2C   NaN
TFAP2D   NaN
TFAP2E   NaN
          ..
BATF3    NaN
CREB1    NaN
TP53     NaN
TP63     NaN
TP73     NaN
Length: 1080, dtype: float64

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<AxesSubplot:xlabel='[TF] predicted from ENCODE RNA', ylabel='Gygi_K562'>

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<AxesSubplot:xlabel='[TF] predicted from ENCODE RNA', ylabel='Mann_K562'>

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<AxesSubplot:xlabel='Gygi_K562', ylabel='Mann_K562'>

	$\Delta G$	$\kon$	$\koff$	$t_{1/2}$
$k_d$
1	12.757685	1e5 / (M * s)	0.0001	0 days 01:55:31.471805599
10	11.340164	1e5 / (M * s)	0.0010	0 days 00:11:33.147180560
100	9.922644	1e5 / (M * s)	0.0100	0 days 00:01:09.314718056
1000	8.505123	1e5 / (M * s)	0.1000	0 days 00:00:06.931471806

	chrom	start	end	name	score	strand	TFs	k562_chip_signal_files	max_signal	p	log10p	max_chip_peak_height	TF_RNA	score_x_RNA	log10p_x_RNA	c	chip_signalBinary	pwm_scoreBinary
0	chr11	47378177	47378186	M10306_2.00	5.672115	-	IKZF1	ENCFF957MXZ	2.06838	0.000502	3.299445	NaN	36.913654	9.280697	121.794567	g	1	1
1	chr11	47378177	47378186	M10306_2.00	5.672115	-	IKZF1	ENCFF053QLR	1.06389	0.000502	3.299445	NaN	36.913654	9.280697	121.794567	g	1	1
2	chr11	47378177	47378191	M04801_2.00	7.176316	+	ETV5	ENCFF042WYK	1.48301	0.000069	4.161535	39.50612	9.482606	9.425776	39.462201	g	1	1
3	chr11	47378177	47378191	M09030_2.00	4.500186	+	E2F1	ENCFF387AXO	0.68479	0.000608	3.216315	125.52399	8.737605	6.667822	28.102885	g	1	1
4	chr11	47378177	47378191	M09030_2.00	4.500186	+	E2F1	ENCFF513XPM	0.91216	0.000608	3.216315	125.52399	8.737605	6.667822	28.102885	g	1	1
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
12235	chr11	47378961	47378972	M04401_2.00	3.655551	-	KLF6	ENCFF955XUI	2.61929	0.000232	3.635227	NaN	5.719905	5.399503	20.793153	g	1	1
12236	chr11	47378961	47378973	M08281_2.00	0.209253	-	ZNF586	ENCFF977XAH	1.42771	0.000318	3.497270	NaN	5.334281	1.883407	18.655423	g	1	1
12237	chr11	47378962	47378975	M09377_2.00	5.301170	-	SMAD4	ENCFF836SJA	3.05519	0.000461	3.336172	119.26441	19.787000	8.286195	66.012841	g	1	1
12238	chr11	47378965	47378974	M03476_2.00	6.107840	+	NFIX	ENCFF431LIE	2.00159	0.000549	3.260517	NaN	3.746789	7.428739	12.216468	g	1	1
12239	chr11	47378965	47378974	M03477_2.00	5.603724	+	NFIX	ENCFF431LIE	2.00159	0.000844	3.073869	NaN	3.746789	6.924623	11.517139	g	1	1

	[TF] predicted from ENCODE RNA	Gygi_K562
TFAP2A	0.519441	-2.802736
TFAP2B	0.121646	-2.548439
TFAP2C	0.155407	-3.410715
TFAP2D	0.000000	NaN
TFAP2E	0.187127	NaN
...	...	...
BATF3	3.407311	NaN
CREB1	80.409642	0.484013
TP53	14.273234	-3.214319
TP63	4.264729	-0.861330
TP73	3.747720	NaN